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\title[ch05]{Chapter 05: Modules Over the Weyl Algebra}
\author[]{SCC LQW}
%\institute[XX大学]{XX大学\quad 数学与统计学院\quad 数学与应用数学专业}
%\date{2025年6月}

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% 封面页
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  \titlepage
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% 目录页
\begin{frame}{Contents}
  \tableofcontents
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% Section 1
\section{THE POLYNOMIAL RING.}
%---------------------------------------------------
\begin{frame}{1.1 LEMMA.}

Let $R$ be a ring and $M$ an irreducible left $R$-module.
\begin{enumerate}
    \item If $0 \neq u \in M$, then $M \cong R/\text{ann}_R(u)$.
    \item If $R$ is not a division ring, then $M$ is a torsion module.
\end{enumerate}

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%---------------------------------------------------
\begin{frame}{1.2 PROPOSITION.}

The $A_n$-module $K[X]$ is an irreducible and torsion $A_n$-module. Besides this,
$$
K[X] \cong A_n / \sum_{1}^{n} A_n \partial_i. 
$$

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%---------------------------------------------------
\begin{frame}{1.3 EXAMPLE.}

Let $g_1, \ldots, g_n \in K[X]$ be polynomials in $x_1,\cdots,x_n$. 

Let $J$ be the left ideal of $A_n$ generated by $\partial_1 - g_1, \ldots, \partial_n - g_n$. 

Every element of $A_n$ is of the form $f + P$, for $f \in K[X]$ and $P \in J$. 

Hence the map $$\psi : A_n / J \longrightarrow K[X]$$ defined by $\psi(f + J) = f$ is an isomorphism of $K$-vector spaces. 

Although the action of the $x$'s is preserved under this isomorphism, it is not an isomorphism of $A_n$-modules. 

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% Section 2
\section{TWISTING.}
%---------------------------------------------------
\begin{frame}{2.1 PROPOSITION.}

Let $R$ be a ring, $M$ a left $R$-module and $\sigma$ an automorphism of $R$. Then:
\begin{enumerate}
    \item $M_\sigma$ is irreducible if and only if $M$ is irreducible.
    \item $M_\sigma$ is a torsion module if and only if $M$ is a torsion module.
    \item If $N$ is a submodule of $M$ then $(M/N)_\sigma \cong M_\sigma / N_\sigma$.
    \item Let $J$ be a left ideal of $R$. Set $\sigma(J) = \{\sigma(r) : r \in J\}$. Then $\sigma(J)$ is a left ideal of $A_n$ and $(R/J)_\sigma \cong R/\sigma^{-1}(J)$.
\end{enumerate}

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\begin{frame}{2.2 PROPOSITION. }

Let $\mathcal{F}$ be the automorphism of $A_n$ defined by $\mathcal{F}(x_i) = \partial_i$ and $\mathcal{F}(\partial_i) = -x_i$. 

Let $M$ be a left $A_n$-module. 

The twisted module $M_{\mathcal{F}}$ is called the Fourier transform of $M$. 

The reason for the name is clear, $\mathcal{F}$ transforms a differential operator with constant coefficients into a polynomial.

This Proposition says that the Fourier transform of $K[X]$ is $K[\partial]$.

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\begin{frame}{2.3 THEOREM. }
For every positive integer $r$ let $\sigma_r$ be the automorphism of $A_n$ which satisfies $\sigma_r(x_i) = x_i$ and $\sigma_r(\partial_i) = \partial_i - x_i^r$.

The modules $K[X]_{\sigma_r}$ form an infinite family of pairwise non-isomorphic irreducible modules over $A_n$.


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% Section 3
\section{HOLOMORPHIC FUNCTIONS.}
%---------------------------------------------------
\begin{frame}{3.1 LEMMA.}

Let $h(z)$ be the holomorphic function $\exp(\exp(z))$. 

For every positive integer $m$ there exists a polynomial $F_m(x) \in \mathbb{C}[x]$ of degree $m$ such that

$$
 d^m h / dz^m = F_m(e^z) h(z). 
$$

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%---------------------------------------------------
\begin{frame}{3.2 PROPOSITION. }

The function $h(z) = \exp(\exp(z))$ is not a torsion element of the $A_1(\mathbb{C})$-module $\mathcal{H}(U)$.

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% Section 4
\section{EXERCISES.}

%---------------------------------------------------
\begin{frame}{EXERCISE 1}
% E 1
%Exercise 4.1. 

Let $g_1, \ldots, g_n$ be polynomials in $K[X]$.
\begin{enumerate}
    \item Show, by induction on $m$, that $\partial_i^m$ may be written in the form $D(\partial_i - g_i) + f$, where $D \in A_n$ and $f \in K[X]$.
    \item Conclude that every element of $A_n$ can be put in the form $Q + f$, where $$Q \in \sum_{i=1}^{n} A_n (\partial_i - g_i)$$ and $f \in K[X]$.
\end{enumerate}

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%---------------------------------------------------
\begin{frame}{EXERCISE 2}
% E 2
%Exercise 4.2. 

Show that if $g_i \in K[x_i]$ then the module $$A_n / \sum_{i=1}^{n} A_n (\partial_i - g_i)$$ is irreducible.

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%---------------------------------------------------
\begin{frame}{EXERCISE 3}
% E 3
%Exercise 4.3. 

A left $R$-module $M$ is cyclic if $M = R \cdot u$, for some $u \in M$. 

Show that an irreducible module is always cyclic.

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%---------------------------------------------------
\begin{frame}{EXERCISE 4}
% E 4
%Exercise 4.4. 

Let $R$ be a simple ring that is not a division ring. 
Suppose that $M, M{\,}'$ are non-zero left torsion $R$-modules. 
Show that if $M$ is cyclic and $M{\,}'$ is irreducible, 
then $M \oplus M{\,}'$ is cyclic.

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Hint: Let $u$ be a generator of $M$. 
Choose $0 \neq a \in R$ such that $au = 0$. 
Since $R$ is simple $aM{\,}' \neq 0$. 
Let $v \in M{\,}'$ with $av \neq 0$. 
Then $u + v$ generates $M \oplus M{\,}'$.


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%---------------------------------------------------
\begin{frame}{EXERCISE 5}
% E 5
%Exercise 4.5. 

Using Exercise 4, show that the direct sum of any finite number of irreducible $A_n$-modules is cyclic.

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%---------------------------------------------------
\begin{frame}{EXERCISE 6}
% E 6
%Exercise 4.6. 

Let $M, M{\,}'$ be left $R$-modules and let $\sigma$ be an automorphism of $R$. 

Show that $(M \oplus M{\,}')_\sigma \cong M_\sigma \oplus M{\,}'_\sigma$.

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%---------------------------------------------------
\begin{frame}{EXERCISE 7}
% E 7
%Exercise 4.7. 
Let $U$ be an open subset of $\mathbb{C}$. 

The set $\mathcal{H}(U)$ of holomorphic functions defined on $U$ contains the polynomial ring $\mathbb{C}[z]$. 

We will make it into a left $A_1(\mathbb{C})$-module. 

Show that $\mathcal{H}(U)$ is not a cyclic $A_1(\mathbb{C})$-module.

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Hint: Let $h \in \mathcal{H}(U)$ be a generator. 
Show that $h$ cannot be constant and that $\exp(h) \notin A_1(\mathbb{C}) \cdot h$.



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%---------------------------------------------------
\begin{frame}{EXERCISE 8}
% E 8
%Exercise 4.8. 

Let $U$ be an open set of $\mathbb{R}^n$. 

Let $C^\infty(U)$ be the real vector space of all functions of class $C^\infty$ defined on $U$. 

Let $x_i$ act by multiplication and $\partial_i$ by differentiation on $C^\infty(U)$. 

Show that this makes $C^\infty(U)$ into a left $A_n(\mathbb{R})$-module. 

Is $C^\infty(U)$ an irreducible $A_n(\mathbb{R})$-module? 

Is it a torsion module? 

Is it cyclic?

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%---------------------------------------------------
\begin{frame}{EXERCISE 9}
% E 9
%Exercise 4.9. 

Are $\sin(e^z)$ and $\cos(e^z)$ torsion elements of the $A_1(\mathbb{C})$-module $\mathcal{H}(\mathbb{C})$?

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%---------------------------------------------------
\begin{frame}{EXERCISE 10}
% E 10
%Exercise 4.10. 

Are $\exp(\sin z)$ and $\exp(\cos z)$ torsion elements of the $A_1(\mathbb{C})$-module $\mathcal{H}(\mathbb{C})$?

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%---------------------------------------------------
\begin{frame}{EXERCISE 11}
% E 11
%Exercise 4.11. 

Show that $\cos z$ and $\sin z$ are torsion elements of $\mathcal{H}(\mathbb{C})$.

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%---------------------------------------------------
\begin{frame}{EXERCISE 12}
% E 12
%Exercise 4.12. 

Let $U$ be the open set $\mathbb{C} \setminus (-\infty, 0]$. 

If $\alpha \in \mathbb{R}$, show that $z{\,}^\alpha$ is a torsion element of the $A_1(\mathbb{C})$-module $\mathcal{H}(U)$.

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%\begin{frame}[allowframebreaks]{REFERENCES}
\begin{frame}{REFERENCES}

\begin{thebibliography}{99}

\bibitem{coutinho} S. C. Coutinho. A Primer of Algebraic D-modules. 

\bibitem{stafford} J. T. Stafford. Module Structure of Weyl Algebras. 

\bibitem{mc-robson} J. C. McConnell and J. C. Robson. Noncommutative Noetherian Rings.  


\end{thebibliography}

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\end{document}
